Shear-horizontal transverse-electric seismoelectric waves in cylindrical double layer porous media*

Wei-Hao Wang(王偉豪), Xiao-Yan Zhu(朱曉焱), Jin-Xia Liu(劉金霞), and Zhi-Wen Cui(崔志文)

Department of Acoustics and Microwave Physics,College of Physics,Jilin University,Changchun 130012,China

Keywords: seismoelectric effect,shear-horizontal and transverse-electric waves,cylindrical layer,porous media

1. Introduction

The reservoirs of natural resources such as oil,water,and natural gases are mostly porous media. The seismoelectric effect related to the electric double layer in a fluid-containing porous medium has potential applications in many fields,e.g.,resource exploration. Pride[1]derived the governing equations that control the coupled electromagnetic-seismic wave propagation. Since then, research on the seismoelectric effect has been widely carried out. Logging is one of the important means to obtain the formation information. Some scholars put forward the method of seismoelectric effect logging based on the experience of acoustic logging. This method receives both acoustic fields and electric fields,which can make up for the lack of information of a single type of wave field. In the seismoelectric logging method, the source and receivers are placed in the borehole, and not influenced by the detection depth.[2]Thus,the application prospect of the method in deep reservoir exploration is better. Hu et al.[3,4]carried out the earliest seismoelertric effect logging simulation. They discovered the electromagnetic head waves propagating at the formation electromagnetic wave velocity,which are independent of the acoustic field and reach the receivers almost simultaneously. In their follow-up work,[5]they proposed a simplified algorithm for calculating the inducing electric fields. Guan et al.[6]gave the rationality proof and applicable conditions for applying the simplified algorithm to calculate the borehole seismoelectric wave fields. Zyserman et al.[7]applied the method of surface seismoelectric logging exploration to study the problems of CO2storage and fluid seepage monitoring in the shallow surface. Guan et al.[8]used the finitedifference method to simulate the seismoelectric wave fields,the results are basically consistent with the simulation results obtained by real axis integral. When the formation outside the borehole is discontinuous, the characteristics of the seismoelectric coupling wave fields and the interface response law of the inducing electric fields are also important research directions. Ding et al.[9]simulated the borehole seismoelectric wave fields in a cylindrical layered formation with discontinuous salinity, the results show that an interface response will occur when the acoustic wave passes through the salinity difference interface. Zhao et al.[10]simulated the seismoelectric wave fields excited by a point acoustic source of a cylindrical double layer porous medium outside the borehole. Gao et al.[11]studied the propagation law of the acoustic-electric coupling waves in the porous elastic hollow cylinder in the fluid.Liu et al.[12]studied the seismoelectric coupling wave fields under the condition of logging while drilling with a cylindrical layered porous medium. The results show that the interface response will occur when the acoustic waves pass through the discontinuous formation. The more obvious the contrast of the elastic or electrical properties on interfaces in the layered porous medium,the stronger the interface response. The above studies are focused on the full-wave analysis of the seismoelectric coupling wave fields. The full wave waveform of the electric field becomes more complicated when the geological structure outside the borehole is complicated. It is difficult to insight the interface response law of the electric field only from the full waveform. Component wave analysis of the seismoelectric wave fields is more helpful to reveal the relationship between the formation characteristics and the wave fields. Hu[13]used the method of secant integral to calculate the electromagnetic head wave. Wang et al.[14]conducted a comprehensive component wave analysis of the borehole seismoelectric wave fields excited by a point source. Wang et al.[15]studied the generation mechanism of the interface converted electromagnetic waves of a cylindrical layered porous formation. It was found that every time the formation fast pwave and shear-horizontal-wave impinge the interface in the porous medium,an interface converted electromagnetic wave response will generate.In this paper,we also use the secant integral method to simulate the interface converted electromagnetic waves, and discuss the interface response law of shearhorizontal and transverse-electric(SH-TE)seismoelectric logging in a cylindrical double-layer porous medium.

The above research can well explain the characteristics of the borehole seismoelectric coupling wave fields excited by several typical acoustic sources. Due to the simple and easy analysis of the SH wave waveforms excited by the shear source, the work of shallow surface exploration using SH waves and the coupling electric fields has been widely carried out in recent years. Zyserman et al.[16]concluded that the signal-to-noise ratio of a shear wave source may be higher than that of a compressional source. Monachesi et al.[17]studied the seismoelectric response law of the SH waves excited by a surface shear source when passing through the seepage zone. In their subsequent work,[18]they studied the response law of the seismoelectric coupled waves excited by the horizontal shear source of the glacial environment. Gao et al.[19]used the finite difference method to simulate the 2D SH-TE seismoelectric waves. Munch et al.[20]investigated the possibility of using SH-TE seismoelectric coupling waves to detect non-aqueous fluid pollution in porous formations. In addition,related research has also shown that the borehole SH waves excited by the annular shear source have certain potential applications in formation exploration. White[21]first proposed the use of an annular shear source attached to the borehole wall for logging, and calculated the situation of a homogeneous medium outside the borehole. When it is a cylindrical layered medium outside the borehole,the characteristics of the acoustic field are different from the case of a homogeneous infinite medium.Yao et al.[22]studied the SH wave fields of a cylindrical double layer elastic medium outside the borehole. The results show that the wave field consists of the critically refracted SH wave and multi-order guided waves(i.e.,cylindrical Love waves).Unlike the Love waves of the semi-infinite plate structure, the dispersion curve of each order Love wave has a cutoff frequency,which is caused by the bending of the interface.The Love wave phase velocity at the cut-off frequency is the SH wave velocity of the outer medium. When the frequency tends to infinity, the phase velocity of each order Love wave tends to the SH wave velocity of the inner medium. That is,the Love wave phase velocity range is Vs1＜VLove＜Vs2,where VLoveis the phase velocity of the multi-mode Love waves,Vs1and Vs2are the SH wave velocities of the inner and the outer medium. In the cylindrical layered porous medium, the dispersion characteristics of Love waves are similar to those in an elastic medium,but the dispersion characteristics are affected by pore parameters.[23]Recent studies[24]have also shown that the circumferential guided waves in a porous cylinder are multi-order and dispersive. Relevant studies indicated that the seismoelectric effect logging method using the borehole SHTE coupling waves also has certain potential applications in exploration. Cui et al.[25]simulated the borehole SH-TE coupling waves of an infinitely homogeneous saturated porous medium. When the formation outside the borehole is homogeneous, the acoustic field has only one component, namely the SH wave. The electric field has two components, which are the interface converted electromagnetic wave and the accompanying electric field of the SH wave. The interface converted electromagnetic waves propagate at the velocity of the formation electromagnetic waves,and the velocities of the accompanying electric fields and the SH waves are consistent.The presence of the borehole-side interface makes the elastic and electrical properties of the porous medium discontinuous.For this scenario,the formation can be regarded as a cylindrical layered porous medium. In order to analyze the borehole SH-TE coupling wave field characteristics of this situation and study the application potential in exploration,we construct an SH-TE acoustic-electric effect logging model for a cylindrical double layer porous medium in this study. The expressions of the basic field quantities inside and outside the borehole are derived,and the real axis integral method is used to simulate the time-domain waveforms. The wave field composition is also analyzed. Finally, in order to study the interface response law when the porous medium outside the borehole has an interface, we adopt the secant integral method to simulate the interface converted electromagnetic waves and analyze the causes of each component.

2. SH-TE seismoelectric waves

In this section, we first introduce the Pride governing equations and solve the complete coupling equations to obtain the expressions of the basic field quantities inside and outside the borehole.Then the real axis integral method is used to simulate the borehole SH-TE coupling wavefields of a cylindrical double layer porous medium and the wave field composition is also analyzed.

2.1. Expressions of the field quantities

Pride[1]combined Biot’s[26]pore elastic wave equations and Maxwell’s electromagnetic equations to derive the coupled acoustic-electromagnetic equations. The electrokinetic coupling effect is controlled by the following two equations:

where J is the current density,E is the electric field strength;σ(ω),κ(ω)and L(ω)are the dynamic conductivity,the dynamic permeability and the electrokinetic coupling coefficient,respectively,which are all the functions of frequency;ρfis the pore fluid density, η is the viscosity coefficient of the pore fluid, u is the solid phase displacement; p is the pore fluid pressure, and w is the seepage displacement. Equation (1)shows that solid-phase displacement acceleration and pressure gradient can generate current, and the electric field can cause seepage. When the electrokinetic coupling coefficient L(ω)is zero,the acoustic field is decoupled from the electric field,and the Pride governing equations will degenerate into Biot’s pore elastic wave equations and Maxwell’s electromagnetic equations.

Fig.1. Schematic diagram of SH-TE seismioelectric effect logging.

Solving the above complete Pride governing equations,the frequency-wavenumber domain expressions of the basic field quantities can be obtained. We study the SH-TE seismoelectric effect logging problem of a cylindrical double-layered porous medium. The model schematic diagram is shown in Fig. 1. The radius of the borehole is a = 0.1 m, and the distance between the interface of the porous medium and the borehole axis is b=1.5 m. There is an annular shear acoustic source attached to the borehole. The acoustic receivers are placed on the borehole wall and the electromagnetic receivers are placed in the borehole fluid. The radial distance of the electromagnetic receivers from the borehole axis is half the borehole radius. Porous medium 1 is a finite thickness porous medium,and porous medium 2 is infinite.

Since the acoustic source is an annular shear source, the axial electric field cannot be excited. According to Cui et al.,[25]the frequency-wavenumber domain expressions of the basic field quantities of the borehole fluid and the outermost infinite porous medium are given by

2.2. Numerical simulation results

In this part,the real axis integral method is used to simulate the coupled acoustic and electromagnetic wave fields.The parameters of the layered porous medium are given in Table 1,they are the typical sandstone parameters.[4,16]In addition,the fluid density in the borehole is 1000 kg/m3, and the salinity is 0.01 Mol/L.The central frequency of the acoustic source is 10 kHz.

Table 1. Porous medium parameters.

Fig.2.Borehole SH-TE seismoelectric wave fields of a cylindrical double layer porous medium. The solid and dashed lines correspond to the electric and acoustic fields,respectively.

Compared with the homogeneous medium,[25]when the porous medium outside the borehole is a cylindrical double layer porous medium, the components of the acoustic field and the electric field become complicated. Due to Vs1＜Vs2,the cylindrical Love waves can be excited.[22,23]The acoustic field is composed of two parts, which are represented as c–c,d–d in Fig. 2, respectively. The component c–c is the critically refracted SH wave, it propagates at the SH wave velocity of the outer porous medium. The component d–d is composed of multi-order Love waves. The electric field consists of four components. When the source distance increases, the arrival time of components a–a and b–b is almost unchanged.This shows that components a–a and b–b are the interface responses,they are generated by the SH waves impinging the interface of the medium and propagate at the velocity of the formation electromagnetic waves, also called the interface converted electromagnetic waves. This is consistent with the interface response characteristics of the borehole seismoelectric coupling wave fields excited by a point acoustic source.[3–5]After the interface responses a–a and b–b, the electric field component c–c is the accompanying electric field of the SH wave. The electric field component d–d is the accompanying electric field of the multi-mode Love waves. Since the propagating velocity of the accompanying electric fields of the Love waves are the same as the Love wave velocities,[3,4,25]the dispersion law for them is also the same. The dispersion characteristics of the Love waves in a cylindrical layered porous medium can refer to Ref.[23]. In a nutshell,as the frequency increases, the phase velocity of the Love wave gradually decreases, and the phase velocity of each order Love wave is greater than the SH wave velocity of the inner layer porous medium and less than the SH wave velocity of the outer layer porous medium.

3. Interface response

In order to study the interface response of the borehole SH-TE seismoelectric coupling wave fields when the porous medium has an interface,we use the secant integral method to calculate the interface converted electromagnetic waves. The interface converted electromagnetic waves refer to the electromagnetic wave components generated by the SH wave impinging the interface of the medium and propagate at the velocity of the formation electromagnetic wave.[13,25]In this section, we first introduce the basic theory of the vertical secant integral and simulate the time domain waveforms of the interface converted electromagnetic waves. Next, the cause of each component is analyzed by comparing the interface converted electromagnetic wave waveforms of different interface positions.

3.1. The secant integral theory

Table 2. Values of the formation body wave branch points.

Fig. 3. Distribution of the body wave branch points and the integral path.

3.2. Interface converted electromagnetic wave

Figure 4 shows the normalized time domain waveforms of the interface converted electromagnetic wave when the formation outside the borehole is a cylindrical double layer porous medium. The parameters adopted in this section are the same as those in Table 1. The distance between the interface of the porous medium and the borehole axis is still 1.5 m,and the axial distance of the electromagnetic sensors from the borehole axis is half the radius of the borehole. In order to distinguish it from the tangential electric field full wave Eθobtained in the previous section,we represent the interface converted electromagnetic waves calculated by secant integral as Eθcut.

The interface converted electromagnetic wave is composed of three components,which are represented as a–a,b–b,and c–c in Fig.4.The arrival time of each component of the interface converted electromagnetic wave hardly increases with the increasing source distance. In order to illustrate the component corresponding relationship between the interface converted electromagnetic wave Eθcutand the electric field full wave Eθ,Fig.5 shows the waveform comparison of Eθcutand Eθ. Here we take the waveforms with a source distance of 3 m as an example. The upper figure is the waveform of the electric field full wave and the lower one is the waveform of the interface converted electromagnetic. They are both normalized with their respective maximum amplitudes.

Fig. 4. Normalized amplitudes of the interface converted electromagnetic wave.

Fig.5. Waveform comparison of the interface converted electric wave and the electric field full wave.

From comparison, we can find that the first two components of the interface electromagnetic wave Eθcutand the electric field full wave Eθare correspondence. This further illustrates that the components a–a and b–b in the full wave of the electric field given by Fig. 2 are the interface converted electromagnetic waves. Since the amplitude of the interface converted electromagnetic wave component c–c in Fig. 4 is small and the arrival time is close to the accompanying electric fields,it is not obvious in the electric field full wave Eθ.In order to explain the production causes of each component of Eθcut,we compare the waveforms of Eθcutof different porous medium interface positions, as shown in Fig. 6. Δt1and Δt2are the arrival time difference, b is the distance between the porous medium interface and the borehole axis. Here we still take the waveforms with a source distance of 3 m as an example.

Fig.6. Interface converted electromagnetic waveforms of different interface positions.The solid and the dashed line correspond to b=1.5 m and 1.8 m,respectively.

As shown in Fig. 6, when the porous medium interface turns 0.3 m away from the borehole wall,the arrival time and the amplitude of the first component hardly change. The position of the interface has little influence on the first component. This shows that the first component(represented as a–a in Fig. 4) is the interface response at the borehole wall. The arrival time difference of the second component is 0.218 ms,which is consistent with the time of SH wave propagating 0.3 m in the inner porous medium. This shows that the second component(represented as b–b Fig.4)is the interface response generated by the SH wave impinging the interface of the porous medium. The arrival time difference of the third component is 0.654 ms, which is consistent with the time of SH wave propagating 0.9 m in the inner medium. This shows that the third component (represented as c–c in Fig. 4) is the interface response generated by the SH wave impinging the interface of the porous medium again after two reflections.Since the SH wave is uncoupled with the fluid in the borehole,there is almost no interface response when the SH wave propagates back from the formation to the borehole wall. In summary,the first component of the interface converted electromagnetic wave is the interface response at the borehole wall. Except for the first component,all the other components are the interface responses generated by the SH wave impinging the interface between the porous medium.

4. Conclusions

In summary, we have studied the borehole SH-TE seismoelectric coupling wave fields of a cylindrical double layer porous medium. First, we derive the expressions of the basic field quantities inside and outside the borehole,and the full wave waveforms of the acoustic field and electric field are simulated. Then,in order to study the interface response,we use the secant integral method to calculate the interface converted electromagnetic waves. The results show that the interface of the porous medium makes the composition of the acousticelectric coupling wave fields more complicated than that of a homogeneous infinite medium. In addition to the critically refracted SH waves,multi-order and dispersive cylindrical Love waves can also be excited due to the layered structure. The electric field includes the interface responses and the accompanying electric fields of the acoustic fields.The interface converted electromagnetic waves propagate in the porous medium at the velocity of the formation electromagnetic waves while the accompanying electric fields are consistent with the propagation velocity of the acoustic waves. The existence of the interface of the porous medium makes the interface response law of the SH-TE wave fields complicated. The results show that when the SH wave generate multi refractions and reflections in the inner porous medium, each time it impinges the interface between the porous medium, an interface converted electromagnetic wave response will occur. The research results in this study indicate that the borehole SH-TE seismoelectric coupling wave fields have certain application potential in the fields of formation exploration and borehole-side interface detection. However, the Love waves in the porous layered medium are complicated. The dispersion,excitation and attenuation characteristics of the cylindrical Love modes with seismoelectric effect are worthy of further study. In addition,the idealized model in this paper needs further improvement to be more in line with the actual situation.

5. Appendix A

The elements of matrix[mij]7×7are

Here a is the borehole radius,and b is the distance between the interface of the porous medium and the borehole axis.